#9065845

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5098 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2,5,6)(3,4,8,7)(11,15)(13,16)(17,18,22,23)(19,27,29,24)(20,25,30,28)(21,31,32,26);(1,2,5,6)(3,7,8,4)(9,10,12,14)(11,13,15,16)(18,23)(20,30)(24,27)(26,31);(2,6)(3,8)(9,11)(10,13)(12,15)(14,16)(18,23)(20,30)(24,27)(26,31);(3,8)(4,7)(17,19)(18,24)(20,31)(21,25)(22,29)(23,27)(26,30)(28,32);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,20)(18,25)(19,26)(21,27)(22,30)(23,28)(24,32)(29,31);(1,4)(2,3)(5,7)(6,8)(9,11)(10,13)(12,15)(14,16)(17,21,22,32)(18,26,23,31)(19,28,29,25)(20,27,30,24)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,e^d=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8973041);